Modular Invariance of Finite Size Corrections and a Vortex Critical Phase

TL;DR

This paper derives exact finite size corrections for a Gaussian model on a toroidal lattice, revealing a vortex critical phase with modular invariant properties and non-monotonic behavior of a cylinder charge as a function of mass.

Contribution

It provides the exact finite size and lattice corrections for a Gaussian model with phases, uncovering a vortex critical phase and demonstrating modular invariance of corrections.

Findings

Finite size corrections are exact and modular invariant.

A vortex critical phase exists at zero mass with non-zero phases.

The cylinder charge varies non-monotonically with mass.

Abstract

We analyze a continuous spin Gaussian model on a toroidal triangular lattice with periods $L_0$ and $L_1$ where the spins carry a representation of the fundamental group of the torus labeled by phases $u_0$ and $u_1$. We find the {\it exact finite size and lattice corrections}, to the partition function $Z$, for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over phases gives the corresponding result for the Ising model. The limits $m\rightarrow0$ and $u_i\rightarrow0$ do not commute. With $m=0$ the model exhibits a {\it vortex critical phase} when at least one of the $u_i$ is non-zero. In the continuum or scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are {\it modular invariant} and for the critical phase are given by elliptic theta functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge'' $c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$.